0.0 0.2 0.4 0.6 0.8 1.0
C D F D M ( M
200)
|ˆ a
1( t )
·ˆ u
1( t
0)
|σ = 10 h
−1cMpc
|
a ˆ
1( t )
·u ˆ
2( t
0)
|σ = 10 h
−1cMpc
|
a ˆ
1( t )
·u ˆ
3( t
0)
|σ = 10 h
−1cMpc
0.0 0.2 0.4 0.6 0.8
|
a ˆ
1( t )
·u ˆ
1( t
0)
|0.0 0.2 0.4 0.6 0.8
CDF CG
σ = 10 h
−1cMpc
0.0 0.2 0.4 0.6 0.8
|
a ˆ
1( t )
·u ˆ
2( t
0)
|σ = 10 h
−1cMpc
0.0 0.2 0.4 0.6 0.8 1.0
|
a ˆ
1( t )
·u ˆ
3( t
0)
|σ = 10 h
−1cMpc
24 6 8 10 12
t [Gyr]
Figure 5.10. Cumulative probability distributions of alignment angles between orienta-tions of haloes at each epoch t and eigenvectors of the tidal field at the present epoch t0. Top and Bottom panels show results for dark matter haloes with the enclosed massM200 and for CGs, respectively. Left, mid-dle, and right panels show the position angles ofuˆ1,uˆ2, anduˆ3 relative to a1(t,DM), respectively. Colour scale corresponds to the cosmic time, bluer lines are earlier and redder lines are later. The smoothing scale of the tidal field ofσ= 10h−1cMpc is adopted.
time-independent manner. They are fairly uncorrelated with uˆ2(t0) at the early epochs, but develop weak correlation toward the present epoch. The correlation of CGs against the tidal field are weaker than that of DM haloes, but exhibits qualitatively a similar trend. This is consistent with the fact that 11 and 29 out of our 40 clusters correspond to
“clusters” and “filaments”, respectively, according to the definition in section 5.1.3 (e.g.
Hahnet al., 2007).
Bate et al. (2019) have studied in particular the evolution of alignments of massive elliptical galaxies relative to the tidal field. They find that the alignments are tighter for ˆ
u1 and ˆu3 than foruˆ2, and also that the alignments increase from z= 3 to 0. These two findings are consistent with our results.
5.4 Summary
In this chapter, we explore the origin of alignments between orientations of BCGs and their host DM haloes by tracing their cosmic evolutions. We use the same 40 cluster-sized
DM haloes and CGs in chapter 3 and identify their progenitors at 50 different epochs fromz= 5 to 0. We then fit their shapes and orientations with a triaxial ellipsoid model following Jing & Suto (2002).
While the orientations of both DM haloes and CGs change significantly due to repeated mergers and smooth mass accretion episodes, their relative orientations are well aligned at each epoch even at high redshifts, z > 1. The result is qualitatively consistent with observations of Westet al. (2017), who reported the mutual alignment between orienta-tions of BCGs and clusters even at high redshift, z > 1.3 (t < 5 Gyr). The alignment becomes tighter with cosmic time; the major axes of the CGs and their host DM haloes at present are aligned on average within 30 deg in three dimensional space and∼20 deg in the projected plane. We also compute the eigen-vectors of the tidal field centred at the location of CG in each halo with smoothing scale of 10h−1cMpc. The orientations of the major axes of DM haloes on average follow one of the eigen-vectors of the surrounding tidal field that corresponds to theslowest collapsing (or even stretching) mode, and the alignment with the tidal field also becomes tighter.
A picture of the evolution of the orientations of CGs and DM haloes emerging from our current study is summarized as follows (see Figure 5.11 for a schematic picture).
Even at early epochs, t = 2 Gyr, orientations of the CG and its host DM halo in an individual system exhibit weakly correlation in a statistical sense. The orientations of both the CG and its host DM halo significantly change due to mergers and mass accretion episodes. However, the orientations of the CG and host DM halo change coherently and evolve together toward their current orientations that are more tightly correlated with the surrounding large-scale matter distributionuˆ3(t0) than at early epochs. This implies that the instantaneous alignment between the DM halo and the CG is driven by strong dynamical interactions through repeated mergers and mass accretion episodes.
Finally, the major axes of both galaxy clusters and CGs tend to be aligned with preferred directions of surrounding matter distributions such as filaments. This can be interpreted as the mass accretion episodes happen statistically more often from the directions of surrounding matter distributions. As a result, the alignments become tighter with time and then the strong alignments at present epoch have been generated. Indeed the CG evolves following that of the host DM halo and becomes tightly aligned with each other;
their typical angles are < 30◦ and < 20◦ in the three dimensional space and in the projected plane, respectively, at the present epoch.
The above basic picture is visually illustrated in Figure 5.12. Each panel depicts the simulation box of (100h−1cMpc)3projected along the z-axis of the simulation. The grey scale represents the surface density of DM particles on (1 h−1cMpc)2 cells at z = 1.97 (top), 0.67 (centre) and 0.16 (bottom). Green bars in the left panels and red bars in the right panels indicate the eigen-vectoruˆ3(t) of the tidal field and the major axis ˆa1(t) of CGs projected on each x-y plane, whereas blue bars in all the panels are the projected major axis ˆa1(t) of DM haloes at epochs around the redshift of each panel. The green bars are roughly aligned along the filamentary structure and do not change so much. The blue bars seem to be aligned with the green bars gradually with time, and the tendency
5.4 Summary 85
t
(ii)
(iii)
DM halo
CG
(i) filaments
Figure 5.11. A schematic picture of evolution of alignment between CG and DM halo in a galaxy cluster. (i) Orientations of CG and DM halo are weakly aligned with each other. (ii) Orientations of CG and DM halo are changed significantly by repeated mergers and mass accretion episodes, but the major axes of them remain to be aligned with each other even after the mass accretions.
(iii) Orientations of both CG and DM halo tend to be aligned with preferred directions of the filament.
of the mutual alignment is stronger between the blue and red bars, i.e., DM haloes and CGs.
In this chapter, we present the predicted evolution of alignments between BCGs, DM haloes, and the large-scale structure, which should be confronted with observations. A caveat is that we focused on the evolution of the same halo over the cosmic time whose mass is different at different epochs (see Figure 5.1). Such difference of masses should be taken into account for a fair comparison with observations (Linet al., 2017). The survey result by Hyper Suprime-Cam Survey (Aiharaet al., 2018) would be useful for examining the redshift evolution of the alignment between orientations of BCGs and clusters because it covers a large (∼1000 deg2) and deep (z∼1.1) area (Oguri et al., 2018).
Figure 5.12. Projected mass density fields of DM and the orientations of CGs (red), DM haloes for the enclosed mass of 0.1M200 (blue), and the tidal field eigenvec-torsuˆ3 (green) for early (top panel,z= 4.25-1.16,t= 1.5-5.4 Gyr), middle (middle panel, z = 1.09-0.39, t = 5.6-9.6 Gyr), and late (bottom panel, z= 0.36-0.018,t= 9.8-13.5 Gyr) epoch. In each panel, all the eigenvectors in the redshift range are shown. The size of each panel corresponds to the simulation box size, 100 h−1cMpc. Lengths of lines indicate orientations with respect to the projection, long lines are nearly perpendicular to the line of sight and short lines are nearly parallel to the line of sight, respec-tively. Grey scales correspond to the surface mass density of DM which are computed by the projection of all particles in the simulation box at middle time for each panel t = 1.97 (top), t = 0.67 (middle), t = 0.16 (bottom) Gyr, respectively.
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Chapter 6
Summary and conclusions
In this thesis, we focus on non-sphericity of galaxy clusters and central galaxies (CGs), especially the ellipticity and the orientation, and investigate systematically to what extent they correlate. The new point of this thesis is to conduct a comprehensive study both theoretically and observationally focusing on the non-sphericity of galaxy clusters. Al-though to carry out simulations incorporating baryon physics is essential to theoretically investigate physical quantities related to CGs, such simulations have been difficult for a long time. Since recently both understanding of baryon physics, especially AGN feed-back, and the computational performance, have advanced, simulations involving reliable baryon physics have been conducted. We made theoretical predictions related to the non-sphericity of galaxy clusters using the Horizon-AGN simulation, which is one of the latest simulations, and also tried to understand the structure formation history in the ΛCDM universe. We validate the consistency of the ΛCDM model complementarily to previous probes by comparing the theoretical predictions with our new observation in addition to previous observations. The results obtained in each chapter are summarized below.
Chapter 3
• We measured ellipticities and orientations of 120 galaxy clusters in the Horizon-AGN simulation using mock observational images in visible light, X-ray, and radio wavelength.
• Mean ellipticity values obtained from dark matter (DM), member galaxy distribu-tion, and X-ray are 0.35, 0.5, and 0.2, respectively, which are marginally consistent with the currently available observations.
• Orientations of galaxy clusters measured in the mock images are aligned well with those of CGs in the ΛCDM universe, and the theoretical predictions were also marginally consistent with currently available observations.
Chapter 4
• We obtained a large sample of galaxy clusters by combining three survey data observed by Hubble Space Telescope. We measured ellipticities and orientations for 45 DM haloes using the strong lensing, and those of CGs from high resolution images to provide a new observational constraint.
• From our new observation, we found that orientations of DM haloes and those of their CGs are aligned well with each other,⟨|θSL−θBCG20 |⟩= 22.2±3.9 deg, and that galaxy clusters are on average more aspherical than central galaxies,⟨eDM−e20BCG⟩= 0.11±0.03.
• We obtained theoretical predictions of the ΛCDM model from the Horizon-AGN simulation that ellipticity values of the DM haloes and central galaxies are almost equal on average from galaxy to cluster scales, but the slightly larger masses are, the larger the differences of ellipticities become such that DM haloes become aspherical.
Chapter 5
• We investigated time evolution of orientations and masses of DM haloes and their CGs for 40 galaxy clusters in the Horizon-AGN simulation over the cosmic time from t= 1.5 to 13.5 Gyr.
• Even in the early stage of the universe,t= 1.5 Gyr, orientations of DM haloes and those of their CGs are weakly correlated, and the alignments at each epoch become tighter with cosmic time.
• Orientations of both DM haloes and CGs have significantly changed through evolu-tion rather than being constant, and the changes of orientaevolu-tions are mainly caused by mass accretion episodes.
• We examined the time evolution of orientations of DM haloes and CGs relative to the directions of surrounding matter distributions around galaxy clusters defined by tidal fields. The major axes of both DM haloes and CGs tend to slightly be aligned with the directions of surrounding matter distributions.
Throughout this thesis, we carried out mock observations of simulated galaxy clusters to understand the structure formation history in the ΛCDM universe, and validated the consistency of the ΛCDM model through comparisons of these results with observations in the real Universe. In the ΛCDM universe, DM haloes grow up following the bottom-up structure formation scenario, in which small structures of DM collapse first and larger structures are formed by their mergers. Galaxy clusters, the largest self-gravitational bounding systems in the universe, are still growing objects by repeated mergers under this scenario.
We confirmed that DM haloes of cluster scales are more elongated than those of galaxy scales in the ΛCDM universe (Figure 4.9). This result can be interpreted that galaxy clusters are growing through anisotropic mass accretion episodes whereas DM haloes at galaxy scales formed relatively earlier epochs and settled in virialization. This scale de-pendence is observed; i.e., mean ellipticity values of DM haloes at galaxy scales are on average⟨e⟩ ∼0.3 (e.g. Hoekstraet al., 2004) while those at cluster scales are⟨e⟩ ∼0.5 (e.g.
Oguriet al., 2010, 2012, and our work). The quantitative consistency of these observations (Figure 4.9) indicates the validation of the ΛCDM model.
We found that galaxy clusters in RELICS sample, which likely contains disturbed clus-ters, are more aspherical than those in CLASH sample, which preferentially comprises relaxed clusters. In addition, double peak clusters, which might be during or before
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merger, are more aspherical than those of single peak indicating that shapes of clusters are elongated by such merger events. Since we did not investigate the effects of such sample selection in this thesis, it remains future tasks to investigate ellipticity variations of galaxy clusters before and after mergers in the ΛCDM universe and compare with these observations.
Orientations of galaxy clusters and CGs are well aligned in the ΛCDM universe (Fig-ure 3.11). The alignments might be because both are affected by anisotropic matter distributions such as filaments imprinted in the initial conditions of primordial density fluctuations. We confirmed the effect of the initial conditions by examining preferred di-rections of surrounding matter distributions defined by the tidal field. According to the bottom-up scenario, galaxy clusters and CGs grow up through mass accretion episodes, and the mass accretions change their orientations (Figure 5.7), but the orientations re-main aligned on average at each time (Figure 5.3). The mass accretions take place along directions of filaments which were already imprinted in the early universe (Figures 5.8 and 5.9). The bottom-up structure formation scenario in the ΛCDM universe explains observational facts that orientations of galaxy clusters and those of their CGs are well aligned (Figure 3.12), and that the alignments already existed at 10 billion years before (Westet al., 2017).
The alignment becomes weaker as the mass decreases (Figure 4.12). Since DM haloes of small mass scales collapse quickly and are approaching the physical equilibrium state in the ΛCDM universe, they have forgotten the memory of the initial conditions. The weaker alignments at small mass scales also support that galaxy clusters and CGs are aligned because they still retain the memory of the initial conditions, rather than because of physical processes such as tidal torques. Observationally, the alignment is also weak at galaxy scales (e.g. Okumura & Jing, 2009), and thus the structure formation scenario predicted by the ΛCDM model is consistent with the observations.
Orientations of X-ray surface brightness are well aligned with those of CGs in the ΛCDM universe (Figure 3.11). Since the cluster sample in the Horizon-AGN simulation is a volume limited sample containing both relaxed and disturbed clusters, there is no bias for gas states. However, we found that the distribution of the alignment angles in the simulation is consistent with that from the CLASH observation composed of relaxed clusters (Figure 3.12), suggesting that the alignment is independent of gas states. This result is qualitatively consistent with observational finding by Hashimotoet al.(2008) and also supports the ΛCDM model.
Future large surveys such as HSC and LSST will enable us to validate the ΛCDM model more precisely by comparing our theoretical predictions in this thesis with the observations. In particular, the theoretical predictions of the ΛCDM model about the evolution scenario of the alignments between orientations of galaxy clusters and CGs will be testable by observing the alignments at high redshifts beyond z = 1. Furthermore, non-sphericities of both DM haloes and their BCGs for more galaxy cluster samples with wider mass range will be obtained from high resolution images of HST and future space telescopes such as JWST. Especially, it plays an important role in directly comparing
with theoretical predictions of the ΛCDM model to measure non-sphericities of relatively low mass galaxy clusters, ∼ 1014M⊙, whose masses are comparable to typical clusters in the Horizon-AGN simulation. In addition to the observational updates, cosmological hydrodynamical simulations would be updated such that they have larger box sizes. The future simulations would generate massive galaxy clusters with mass of ∼ 1015M⊙ cor-responding to typical observed clusters. Therefore, our finding that the observed mean value of ellipticity differences between galaxy clusters and BCGs is inconsistent with those of the simulated galaxy clusters would be directly testable by updating both observations and simulations.
Once we accept the validity of the ΛCDM model, we would be able to constrain DM model and AGN feedback, and explore baryon distributions in the Universe from the non-sphericity of galaxy clusters. Since the non-non-sphericity of galaxy clusters is sensitive to the cross section of the self-interacting DM (e.g. Yoshidaet al., 2000b) and the strength of the AGN feedback (e.g. Sutoet al., 2017), these parameters can be constrained by comparing results of simulations with observations. The tendency of galaxy clusters to be aligned with filaments can also be used to search for baryons in filaments. About 30% of baryons in the Universe are missing (Fukugitaet al., 1998) compared with the ΛCDM model and are believed to exist in filaments (de Graaffet al., 2019; Tanimura et al., 2019). Future large surveys will explore these baryons within the filaments to validate the ΛCDM model, and thus to identify locations of filaments should be important. The orientations of galaxy clusters can be used as indicators of the filament locations. In any case, our study in this thesis will serve as a bridge between previous and future studies in terms that we focus on the non-sphericity of galaxy clusters and attempt to extract cosmological and astrophysical informations through comparison of observations and theoretical predictions.
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Appendix A
The Horizon simulation:
cosmological hydrodynamical simulation
We examine the correlations of non-sphericities of projected surface densities among dif-ferent components of simulated galaxy clusters and the evolution of the non-sphericities and orientations of DM haloes and their CGs in this thesis. In particular, we are in-terested in the alignment between the position angles of DM haloes and those of their CGs. Clearly this requires a cosmological hydrodynamical simulation implemented with detailed baryon physics and also with high spatial and mass resolutions to identify CGs in the cluster centres. We thus focus on the Horizon simulation, one of the state-of-the-art cosmological hydrodynamical simulations. The detail of this simulation is already described in Dubois et al. (2014). Thus we summarize only its major features relevant to our current studies. The Horizon simulation adopts the standard ΛCDM cosmological model. The cosmological parameters are based on the seven-year Wilkinson Microwave Anisotropy Probe (Komatsu et al., 2011); Ωm,0= 0.272 (total matter density at present day), ΩΛ,0= 0.728 (dark energy density at present day), Ωb,0= 0.045 (baryon density at present day),σ8= 0.81 (amplitude of the power spectrum of density fluctuations that are averaged on spheres of 8h−1 Mpc radius at present day), H0 = 70.4 km/s/Mpc (Hubble constant), andns = 0.967 (the power-law index of the primordial power spectrum), and thus we use these values throughout this thesis to suit the Horizon simulation.
A.1 Detail of the Horizon simulation: box size, resolution, resolved components, and how to solve them
The simulation is performed in a periodic cube of (100h−1Mpc)3, and the initial condition is generated with MPGRAFIC software (Prunetet al., 2008). The simulation follows the evolution of three different components, dark matter, gas, and star. Dark matter is rep-resented byN = 10243 equal-mass particles in the entire box, corresponding to the mass
resolution of 8.27×107M⊙. Baryon gas is assigned over the meshes in the simulation box, and its evolution is solved with the adaptive mesh refinement code RAMSES (Teyssier, 2002). Star is represented by collisionless particles, whose formation is modeled on the basis of an empirical Schmidt law. Since those star particles are created according to a random Poisson process, their masses are not the same, but typically around 2×106M⊙. The evolution of collisionless particles (dark matter and star) are followed by the particle-mesh solver with a cloud-in-cell interpolation. Therefore, the spatial resolution depends on the size of the local cell where those particles are located. The initial size of the gas cell is 136 kpc, and then refined up to 1.06 kpc (= 136/27kpc after seven times refinement), which corresponds to the highest spatial resolution achieved in the simulation.
In addition to radiative cooling and hydrodynamical evolution of gas component, feed-back from stars is implemented assuming the Salpeter initial mass function (Salpeter, 1955) with lower and upper mass limits of 0.1M⊙ and 100M⊙, respectively. The mechan-ical energy from Type II supernova explosions and stellar winds is computed according to the STARBURST99 (Leithereret al., 1999, 2010) with the frequency of Type Ia supernova explosions computed using Greggio & Renzini (1983).